14. Sum of Degrees of Vertices. The sum of the degrees of the vertices in a graph
with a finite
number of edges is twice the number of edges.
15. Connected. A graph is called connected if there is a p
8. Variance and Standard Deviation for Bernoulli Trials. In Bernoulli trials with
probability
p of success, the variance for one trial is p(1p) and for n trials is np(1p), so the
standard
deviation fo
5. For which values of n does the complete graph on n vertices have an Eulerian
Circuit?
6. The hypercube graph Qn has as its vertex set the n-tuples of zeros and ones. Two
of these
vertices are adjac
If you analyze our proof of Diracs theorem, you will see that we really used only a
consequence
of the condition that all vertices have degree at least v/2, namely that for any two
vertices, the
sum o
been marked with an a and so is in Y A. But every other edge must be covered by
XA
because in a bipartite graph, each edge must be incident with a vertex in each part.
Therefore
C is a vertex cover. I
Tour of the connected component of G_ containing xk+1 to our tour. When we add the
last edge
and vertex of our closed path to the path we have been constructing, every vertex
and edge of the
graph wil
listing all vertices adjacent to it. In the case of multiple edges, we list each
adjacency as many
times as there are edges that give the adjacency. In our pseudocode we implement
the idea of
an adjac
than the number of registers our computer has available. The problem of assigning
variables to
registers is called the register assignment problem.
An intersection graph of a set of intervals of real
nonempty children. Thus removing the root gives us two binary trees, rooted at the
children of
the original root, each with fewer than n vertices. By the definition of full, each of
the subtrees
roote
measure of the size of the input graph, and k is independent of n), including
counting as one step
solving an instance of the first problem, and accepts exactly the instances of the
second problem
tha
components of the graph GW that results from removing the closed walk, and then
follow our
closed walk, pausing each time we enter a new connected component of G W to
recursively
construct an Eulerian
1
3
3
1
3
1
3
11
1
3
3
3
3
1
3
which vertex b is inside the cycle C and one in which it is outside C. (Notice also
that in both
cases, we have more than one choice for the cycle because there are two
5. Find all induced cycles in the graph of Figure 6.9.
6. What is the size of the largest induced Kn in Figure 6.9?
7. Find the largest induced Kn (in words, the largest complete subgraph) you can in
vertex we add to the tree from vertex V [Vertex[m] will have distance n+1 from the
tree. Thus
every vertex added to the tree from a vertex of distance n from V [x] will have
distance n + 1
from V [x].
of connected components wont change. If the endpoints of the edge are in different
connected
components, then the number of connected components can go down by one. Since
an edge has
two endpoints, it
maximum matching, that is, a matching that is at least as big as any other
matching.
The graph in Figure 6.22 is an example of a bipartite graph. A graph is called
bipartite
whenever its vertex set ca
not in S. Thus when it stops, S will be the vertex set of a connected component of
the graph
and E_ will be the edge set of a spanning tree of this connected component. This
suggests that
one use that
NP-complete, 295, 296
NP-complete Problems, 294
number theory, 4081
one-to-one function, 11
one-way function, 68, 70
only if (in logic), 93
onto function, 11
or
exclusive (in logic), 85
or (in logic),
polynomial time and the class NP of problems said to be solvable in
nondeterministic polynomial
time. We are not going to describe these problem classes in their full generality. A
course in formal
la
trials large enough, we can make the probability that the number X of successes is
between
np ns and np + ns as close to 1 as we choose. For example, we can make the
probability
that the number of suc
collision, 187, 191
collisions, 228, 234
empty slots, 228, 234
expected maximum number of keys per
slot, 233, 234
expected number of collisions, 228, 234
expected number of hashes until all slots
occu
13. Prove that a graph is an r-tree, as defined at the end of the section if and only if
it is a
rooted tree.
14. Use the inductive definition of a rooted tree (r-tree) given at the end of the
section
suitable. If you can do so, give an assignment of people to jobs. If you cannot, try to
explain why not.
Exercise 6.4-2 Table 6.2 shows a second sample of the kinds of applications a
school
district m
3. A path that includes each vertex of the graph at least once and each edge of the
graph
exactly once, but has different first and last endpoints, is known as an Eulerian Trail
4. A graph G has an Eu
guarantee a Hamiltonian cycle, because when we got to the second last vertex on
the cycle, all
of the n 2 vertices it is adjacent to might already be on the cycle and different
from the first
vertex,
sure that we do not examine edges that point from vertex i back to a vertex that is
already in
the tree.
This algorithm requires that we execute the For loop that starts in Line 4 once for
each
edge i
15. Draw a graph of the equation y = x(1 x) for x between 0 and 1. What is the
maximum
value of y? Why does this show that the variance (see Problem 10 in this section) of
the
number of successes rand
measure of the size of the input graph, and k is independent of n), including
counting as
one step solving an instance of the first problem, and accepts exactly the instances
of the
second problem tha